Unit 1197 - Mathematical Statistics

1. Introduction to Mathematical Statistics

• Role of statistics in data analysis
• Types of data: qualitative vs. quantitative, discrete vs. continuous
• Key terminologies: population, sample, parameter, statistic, variable

2. Descriptive Statistics

• Measures of central tendency: mean, median, mode
• Measures of dispersion: variance, standard deviation, range, interquartile range
• Graphical representations:
  • Histograms
  • Box plots
  • Stem-and-leaf plots
  • Frequency distributions

3. Probability Distributions

• Overview of probability concepts
• Discrete distributions:
  • Binomial distribution: definition, properties, applications
  • Poisson distribution: definition, properties, applications
• Continuous distributions:
  • Normal distribution: properties, standard normal distribution, empirical rule
  • Uniform distribution: definition and applications

4. Sampling Techniques

• Importance of sampling in statistics
• Sampling methods:
  • Simple random sampling
  • Stratified sampling
  • Cluster sampling
  • Systematic sampling
• Sampling biases and errors

5. Estimation Theory

• Point estimation:
  • Definition and properties (unbiasedness, consistency, efficiency)
  • Common estimators for mean and variance
• Interval estimation:
  • Confidence intervals for population mean and proportion
  • Interpretation of confidence levels
• Methods of estimation: Method of moments, Maximum likelihood estimation (overview)

6. Hypothesis Testing

• Formulating hypotheses: null and alternative
• Significance levels and p-values
• Types of errors: Type I and Type II
• Test statistics and critical values
• Common hypothesis tests:
  • Z-test
  • t-test (one-sample, two-sample)
  • Chi-square test for independence and goodness-of-fit

7. Regression Analysis

• Introduction to regression and correlation
• Simple linear regression:
  • Model formulation
  • Least squares method
  • Interpretation of regression coefficients
• Multiple regression overview
• Assessing model fit:
  • Coefficient of determination (R²)
  • Residual analysis

8. Analysis of Variance (ANOVA)

• Purpose and assumptions of ANOVA
• Sources of variation: between-group and within-group
• One-way ANOVA:
  • Calculations of sums of squares
  • F-test statistic
• Interpreting ANOVA results
• Post-hoc tests (brief overview)

9. Nonparametric Statistics

• When to use nonparametric methods
• Wilcoxon rank-sum test (Mann-Whitney U test)
• Kruskal-Wallis test
• Spearman's rank correlation coefficient

10. Bayesian Statistics

• Introduction to Bayesian reasoning
• Bayes' theorem and its interpretation
• Prior, likelihood, and posterior probabilities
• Bayesian inference concepts
• Applications and advantages of Bayesian methods

Each section includes theory, examples, and practical exercises to reinforce learning.